The Finite Element Method

In practical situations the determination of suitable basis functions for use in the Galerkin method can be extremely difficult, especially in cases for which the domain Ω does not have a simple shape. The finite element method overcomes this difficulty by providing a systematic means for generating basis functions on domains of fairly arbitrary shape. What makes the method especially attractive is the fact that these basis functions are piecewise polynomials that are nonzero only on a relatively small part of Ω, so that computations may be carried out in a modular fashion, which is well suited to computer-based approaches. As we show a little later, the family of spaces V h (h ∈ (0, 1)) defined by the finite element procedure possesses the property that V h approaches V as h approaches zero, in an appropriate sense. This is, of course, an indispensable property for convergence of the Galerkin method.